In a new petition before the court, two Louisiana men who were exonerated after 27 years behind bars are asking the justices to review their case and allow them to sue the New Orleans prosecutor’s office for money damages for violating their constitutional rights. If the justices agree to hear the petition, they could undo some of the harm they did in 2011 and help cure what one federal appellate judge has called an “epidemic” of prosecutorial misconduct across the country.

Earl Truvia and Gregory Bright were convicted of a 1975 murder in New Orleans and sentenced to life. Their conviction was based on the testimony of a single eyewitness, a schizophrenic heroin addict who testified under a false name to hide her criminal history. Local prosecutors did not turn over this and other key evidence to the defendants. In 2002, both men were exonerated largely for this reason. After their release, they sued the office of the longtime New Orleans district attorney, Harry Connick, for letting his prosecutors railroad them into prison.

Under a landmark Supreme Court decision, Brady v. Maryland, prosecutors must disclose any material evidence that could exonerate a defendant. But because individual prosecutors are immune from being sued, the only way to hold the government accountable is if a court finds a systemic failure to train prosecutors properly on the Brady rule.

This would seem to be an easy bar to clear in New Orleans, where, as Mr. Truvia and Mr. Bright argue, Mr. Connick effectively had a policy of not turning over exculpatory evidence. He consistently neglected to provide any such training to his staff, even though the office’s failure to disclose exculpatory evidence led to the exoneration of at least 12 people since 1990. A former assistant prosecutor, they say, described the office’s unwritten policy as “when in doubt, don’t give it up.”

Yet, the United States Court of Appeals for the Fifth Circuit threw out the lawsuit, saying the men had not proved that Mr. Connick’s office had any policy to withhold evidence or that he had failed to train his prosecutors. Nor, it said, had they proved there were any Brady violations before their convictions.

The behavior of Mr. Connick’s office was also at issue in the 2011 Supreme Court case. John Thompson, convicted of murder and armed robbery, spent 18 years behind bars, 14 of those on death row, before a private investigator discovered that several prosecutors in Mr. Connick’s office had lied for years about a crime-lab report that ultimately led to Mr. Thompson’s exoneration.

After Mr. Thompson won a $14 million jury verdict for his suffering, the Supreme Court tossed it out. Justice Clarence Thomas, writing for the majority, said Mr. Thompson had shown only a single Brady violation, not a pattern of misconduct. But as Justice Ruth Bader Ginsburg pointed out in her dissent, the violation was part of the office’s larger culture of disregard for defendants’ basic constitutional rights. Mr. Connick misunderstood the Brady rule so profoundly that he was once indicted himself for suppressing evidence, and he never disciplined a prosecutor for violating the Brady rule.

While New Orleans is among the worst, it’s not alone in violating defendants’ right to exculpatory evidence. Federal and state prosecutors nationwide often fail to honor the Brady rule and are virtually never punished for it. Because Brady violations are by their nature often hidden, one partial fix would be to require prosecutors to turn over their criminal case files to the defense. Ohio and North Carolina have adopted versions of this approach.

The Supreme Court could help Mr. Truvia and Mr. Bright get a measure of justice by giving them a chance to prove their case, and make the government pay when prosecutors ignore the Constitution.

A version of this article appears in print on , on Page A16 of the New York edition with the headline: How to Force Prosecutors to Play Fair. Order Reprints | Today’s Paper | Subscribe